Annotation of src/lib/libm/src/e_j1.c, Revision 1.12.58.1
1.1 jtc 1: /* @(#)e_j1.c 5.1 93/09/24 */
2: /*
3: * ====================================================
4: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5: *
6: * Developed at SunPro, a Sun Microsystems, Inc. business.
7: * Permission to use, copy, modify, and distribute this
1.10 simonb 8: * software is freely granted, provided that this notice
1.1 jtc 9: * is preserved.
10: * ====================================================
11: */
1.3 jtc 12:
1.9 lukem 13: #include <sys/cdefs.h>
1.7 jtc 14: #if defined(LIBM_SCCS) && !defined(lint)
1.12.58.1! bouyer 15: __RCSID("$NetBSD: e_j1.c,v 1.13 2017/02/09 21:23:11 maya Exp $");
1.3 jtc 16: #endif
1.1 jtc 17:
18: /* __ieee754_j1(x), __ieee754_y1(x)
19: * Bessel function of the first and second kinds of order zero.
20: * Method -- j1(x):
21: * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
22: * 2. Reduce x to |x| since j1(x)=-j1(-x), and
23: * for x in (0,2)
24: * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
25: * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
26: * for x in (2,inf)
27: * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
28: * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
29: * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
30: * as follow:
31: * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
32: * = 1/sqrt(2) * (sin(x) - cos(x))
33: * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
34: * = -1/sqrt(2) * (sin(x) + cos(x))
35: * (To avoid cancellation, use
36: * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
37: * to compute the worse one.)
1.10 simonb 38: *
1.1 jtc 39: * 3 Special cases
40: * j1(nan)= nan
41: * j1(0) = 0
42: * j1(inf) = 0
1.10 simonb 43: *
1.1 jtc 44: * Method -- y1(x):
1.10 simonb 45: * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
1.1 jtc 46: * 2. For x<2.
1.10 simonb 47: * Since
1.1 jtc 48: * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
49: * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
50: * We use the following function to approximate y1,
51: * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
52: * where for x in [0,2] (abs err less than 2**-65.89)
53: * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
54: * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
55: * Note: For tiny x, 1/x dominate y1 and hence
56: * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
57: * 3. For x>=2.
58: * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
59: * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
60: * by method mentioned above.
61: */
62:
1.12 drochner 63: #include "namespace.h"
1.5 jtc 64: #include "math.h"
65: #include "math_private.h"
1.4 jtc 66:
1.1 jtc 67: static double pone(double), qone(double);
68:
1.10 simonb 69: static const double
1.1 jtc 70: huge = 1e300,
71: one = 1.0,
72: invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
73: tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
74: /* R0/S0 on [0,2] */
75: r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
76: r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
77: r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
78: r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
79: s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
80: s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
81: s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
82: s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
83: s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
84:
1.5 jtc 85: static const double zero = 0.0;
1.1 jtc 86:
1.11 wiz 87: double
88: __ieee754_j1(double x)
1.1 jtc 89: {
90: double z, s,c,ss,cc,r,u,v,y;
1.6 jtc 91: int32_t hx,ix;
1.1 jtc 92:
1.5 jtc 93: GET_HIGH_WORD(hx,x);
1.1 jtc 94: ix = hx&0x7fffffff;
95: if(ix>=0x7ff00000) return one/x;
96: y = fabs(x);
97: if(ix >= 0x40000000) { /* |x| >= 2.0 */
98: s = sin(y);
99: c = cos(y);
100: ss = -s-c;
101: cc = s-c;
102: if(ix<0x7fe00000) { /* make sure y+y not overflow */
103: z = cos(y+y);
104: if ((s*c)>zero) cc = z/ss;
105: else ss = z/cc;
106: }
107: /*
108: * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
109: * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
110: */
111: if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
112: else {
113: u = pone(y); v = qone(y);
114: z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
115: }
116: if(hx<0) return -z;
117: else return z;
118: }
119: if(ix<0x3e400000) { /* |x|<2**-27 */
120: if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
121: }
122: z = x*x;
123: r = z*(r00+z*(r01+z*(r02+z*r03)));
124: s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
125: r *= x;
126: return(x*0.5+r/s);
127: }
128:
129: static const double U0[5] = {
130: -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
131: 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
132: -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
133: 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
134: -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
135: };
136: static const double V0[5] = {
137: 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
138: 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
139: 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
140: 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
141: 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
142: };
143:
1.11 wiz 144: double
145: __ieee754_y1(double x)
1.1 jtc 146: {
147: double z, s,c,ss,cc,u,v;
1.6 jtc 148: int32_t hx,ix,lx;
1.1 jtc 149:
1.5 jtc 150: EXTRACT_WORDS(hx,lx,x);
1.1 jtc 151: ix = 0x7fffffff&hx;
152: /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
1.10 simonb 153: if(ix>=0x7ff00000) return one/(x+x*x);
1.1 jtc 154: if((ix|lx)==0) return -one/zero;
155: if(hx<0) return zero/zero;
156: if(ix >= 0x40000000) { /* |x| >= 2.0 */
157: s = sin(x);
158: c = cos(x);
159: ss = -s-c;
160: cc = s-c;
161: if(ix<0x7fe00000) { /* make sure x+x not overflow */
162: z = cos(x+x);
163: if ((s*c)>zero) cc = z/ss;
164: else ss = z/cc;
165: }
166: /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
167: * where x0 = x-3pi/4
168: * Better formula:
169: * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
170: * = 1/sqrt(2) * (sin(x) - cos(x))
171: * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
172: * = -1/sqrt(2) * (cos(x) + sin(x))
173: * To avoid cancellation, use
174: * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
175: * to compute the worse one.
176: */
177: if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
178: else {
179: u = pone(x); v = qone(x);
180: z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
181: }
182: return z;
1.10 simonb 183: }
1.1 jtc 184: if(ix<=0x3c900000) { /* x < 2**-54 */
185: return(-tpi/x);
1.10 simonb 186: }
1.1 jtc 187: z = x*x;
188: u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
189: v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
190: return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
191: }
192:
193: /* For x >= 8, the asymptotic expansions of pone is
194: * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
195: * We approximate pone by
196: * pone(x) = 1 + (R/S)
197: * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
198: * S = 1 + ps0*s^2 + ... + ps4*s^10
199: * and
200: * | pone(x)-1-R/S | <= 2 ** ( -60.06)
201: */
202:
203: static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
204: 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
205: 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
206: 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
207: 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
208: 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
209: 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
210: };
211: static const double ps8[5] = {
212: 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
213: 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
214: 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
215: 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
216: 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
217: };
218:
219: static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
220: 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
221: 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
222: 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
223: 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
224: 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
225: 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
226: };
227: static const double ps5[5] = {
228: 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
229: 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
230: 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
231: 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
232: 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
233: };
234:
235: static const double pr3[6] = {
236: 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
237: 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
238: 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
239: 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
240: 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
241: 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
242: };
243: static const double ps3[5] = {
244: 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
245: 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
246: 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
247: 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
248: 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
249: };
250:
251: static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
252: 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
253: 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
254: 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
255: 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
256: 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
257: 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
258: };
259: static const double ps2[5] = {
260: 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
261: 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
262: 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
263: 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
264: 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
265: };
266:
1.11 wiz 267: static double
268: pone(double x)
1.1 jtc 269: {
270: const double *p,*q;
271: double z,r,s;
1.6 jtc 272: int32_t ix;
1.9 lukem 273:
1.5 jtc 274: GET_HIGH_WORD(ix,x);
275: ix &= 0x7fffffff;
1.12.58.1! bouyer 276: if(ix>=0x40200000) {p = pr8; q= ps8;}
! 277: else if(ix>=0x40122E8B) {p = pr5; q= ps5;}
! 278: else if(ix>=0x4006DB6D) {p = pr3; q= ps3;}
! 279: else /*if(ix>=0x40000000)*/{p = pr2; q= ps2;}
1.1 jtc 280: z = one/(x*x);
281: r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
282: s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
283: return one+ r/s;
284: }
1.10 simonb 285:
1.1 jtc 286:
287: /* For x >= 8, the asymptotic expansions of qone is
288: * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
289: * We approximate pone by
290: * qone(x) = s*(0.375 + (R/S))
291: * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
292: * S = 1 + qs1*s^2 + ... + qs6*s^12
293: * and
294: * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
295: */
296:
297: static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
298: 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
299: -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
300: -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
301: -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
302: -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
303: -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
304: };
305: static const double qs8[6] = {
306: 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
307: 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
308: 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
309: 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
310: 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
311: -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
312: };
313:
314: static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
315: -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
316: -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
317: -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
318: -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
319: -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
320: -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
321: };
322: static const double qs5[6] = {
323: 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
324: 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
325: 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
326: 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
327: 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
328: -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
329: };
330:
331: static const double qr3[6] = {
332: -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
333: -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
334: -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
335: -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
336: -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
337: -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
338: };
339: static const double qs3[6] = {
340: 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
341: 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
342: 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
343: 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
344: 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
345: -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
346: };
347:
348: static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
349: -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
350: -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
351: -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
352: -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
353: -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
354: -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
355: };
356: static const double qs2[6] = {
357: 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
358: 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
359: 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
360: 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
361: 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
362: -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
363: };
364:
1.11 wiz 365: static double
366: qone(double x)
1.1 jtc 367: {
368: const double *p,*q;
369: double s,r,z;
1.6 jtc 370: int32_t ix;
1.9 lukem 371:
1.5 jtc 372: GET_HIGH_WORD(ix,x);
373: ix &= 0x7fffffff;
1.12.58.1! bouyer 374: if(ix>=0x40200000) {p = qr8; q= qs8;}
! 375: else if(ix>=0x40122E8B) {p = qr5; q= qs5;}
! 376: else if(ix>=0x4006DB6D) {p = qr3; q= qs3;}
! 377: else /*if(ix>=0x40000000)*/{p = qr2; q= qs2;}
1.1 jtc 378: z = one/(x*x);
379: r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
380: s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
381: return (.375 + r/s)/x;
382: }
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